While these are of course only different ways of writing the exact conservation equation for the water age distribution, they show that, through equation 5, ω(T,t) contains some extra time variability due to the input and output temporal dependence. To address this issue, van der Velde et al. [2012] and Harman [2015] sought different transformations of the SAS. The extra complications, which result from working on PDFs, instead of unnormalized distributions, have been known since the time of Boltzmann [Boltzmann, 1872; Cercignani, 1988], who wrote his transport equation with unnormalized distributions. Thus, the loss functions, n0(t,τ) or μ(T,t), appear naturally when one writes the conservation equation in terms of age mass distribution as in (4) [von Foerster, 1959; Trucco, 1965; Murray, 2002]. In particular, μ(T,t) does not directly contain the time variability of the input and the output (although some time variability is expected when the problem is nonlinear). Thus, for the case of linear, time-invariant system, Eq. 1 and 4, when integrated, simply lead to the classical instantaneous unit hydrograph (IUH) [Sherman, 1932; Dooge, 1959; Nash, 1959]. We note that in the case of multiple outputs (e.g., evapotranspiration and runoff), the use of different loss functions, still allows an input-output analysis as in (7), along with its spectral representation [Priestley, 1981]. This will be presented elsewhere. One would hope to find more interesting developments when the system is nonlinear. In such cases, nonlinearities could be explicitly taken into account in the loss function [Gurtin and MacCamy, 1974] and possibly derived from spatially explicit formulations [Ginn, 1999]. Unfortunately, an always-useful loss function is not available. In general, the μ function is time varying in the nonlinear case, while other formulations are case-specific. For example, the rSAS function [Harman, 2015], in which the loss term is a function of , is time-invariant in the special nonlinear case of plug flow (where the nonlinearity is a pure threshold), but it is time varying even in the simplest case of a well-mixed linear system. As a partial solution one can perhaps use techniques from nonlinear dynamical systems to find locally linear representations [Porporato and Ridolfi, 2003; Kirchner, 2009], or work on low-order moment closures without describing the full distribution [Duffy, 2010]. This work was partially funded by the National Science Foundation through grants CBET-1033467, EAR-1331846, FESD-1338694, and EAR-1316258. We thank C. Harman, L. Ridolfi and G. Katul for useful discussions. This paper is a theoretical study and hence no data were used.
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May 2, 2017
Daniel J Hernández + 2
Open Access
